In this paper we consider the problem of motion of a rigid body in an ideal fluid with two material points moving along circular trajectories. The controllability of this system on the zero level set of first integrals is shown. Elementary “gaits” are presented which allow the realization of the body’s motion from one point to another. The existence of obstacles to a controlled motion of the body along an arbitrary trajectory is pointed out.

On the basis of experimental data of the study of amplitude-dependent internal friction phenomena in polycrystalline solids the inelastic hysteretic state equation with a saturation of nonlinear losses is proposed. Theoretical analysis of the asymmetric saw-tooth waves propagation in such media is carried out. The regularities (amplitude dependent losses and changes in the propagation velocity) are determined for the characteristics of nonlinear wave and its higher harmonic amplitudes. The graphical analysis of form of the wave and evolution of its spectral components is carried out.

The effect of the external measurement noise on characteristics of the Granger causality method was considered for unidirectionally coupled non-linear etalon systems in different oscillation regimes. Coupled maps with the same and different evolution operator in driving and driven systems were studied, as well as coupled flows. The nontrivial dependency of method characteristics was shown in all considered cases for certain parameters and coupling intensity. The reason why this dependency in not monotonous was found out.

The motion of a time-periodic two-degree-of-freedom Hamiltonian system in the neighborhood of the equilibrium being stable in the linear approximation is considered. The weak Raman thirdorder resonance and the strong fourth-order resonance are assumed to occur simultaneously in the system. The behavior of the approximated (model) system is studied in the stability domain of the fourth-order resonance. Areas of the parameters (coefficients of the normalized Hamiltonian) are found for which all motions of the system are bounded if they begin in a sufficiently small neighborhood of the equilibrium. Boundedness domain estimate is obtained. А disturbing effect of the double resonance on the motion of the system within the boundedness domain is described.

Stability of the motion of a thin homogeneous disk in a uniform gravitational field above a fixed horizontal plane is investigated. Collisions between the disk and the plane are assumed to be absolutely elastic, and friction is negligible. In unperturbed motion, the disk rotates at a constant angular velocity about its vertical diameter, and its center of gravity makes periodic oscillations along a fixed vertical as a result of collisions. The stability problem depends on two dimensionless parameters characterizing the magnitude of the angular velocity of the disk and the height of his jump above the plane in the unperturbed motion. An exact solution of the problem of stability is obtained for all physically admissible values of these parameters.

We consider a modified system of two pendulums rods of which intersect and slide without any friction. The pendulums are connected by an elastic linear spring and arranged in a fixed vertical plane of the uniform gravity field. We have shown that there are symmetric and asymmetric equilibrium solutions with respect to the vertical axis. It is revealed that the stability of the model depends on two parameters, the first one specifies the spring stiffness, and the second one defines the distance between the hinges. The conditions of stability and instability of the symmetric equilibrium are obtained in the upper and lower position of pendulums. The analysis of asymmetric equilibrium solutions and stability conditions is carried out for long pendulums. Comparison with the sympathetic pendulums model proposed by Sommerfeld indicates that asymmetric solutions exist only for the modified model.

This paper presents the results of experimental investigations for the rolling of a spherical robot of combined type actuated by an internal wheeled vehicle with rotor on a horizontal plane. The control of spherical robot based on nonholonomic dynamical by means of gaits. We consider the motion of the spherical robot in case of constant control actions, as well as impulse control. A number of experiments have been carried out confirming the importance of rolling friction.

This paper develops topological methods for qualitative analysis of the behavior of nonholonomic dynamical systems. Their application is illustrated by considering a new integrable system of nonholonomic mechanics, called a nonholonomic hinge. Although this system is nonholonomic, it can be represented in Hamiltonian form with a Lie –Poisson bracket of rank 2. This Lie – Poisson bracket is used to perform stability analysis of fixed points. In addition, all possible types of integral manifolds are found and a classification of trajectories on them is presented.

This paper is a review of the problem of the constructive reduction of nonholonomic systems with symmetries. The connection of reduction with the presence of the simplest tensor invariants (first integrals and symmetry fields) is shown. All theoretical constructions are illustrated by examples encountered in applications. In addition, the paper contains a short historical and critical sketch covering the contribution of various researchers to this problem.